Use the remainder theorem

Given the polynomial x4+ax2+bx+cx^4+ax^2+bx+c has a factor (x−2)(x-2) and gives remainders 12 and 26 when divided by (x−3)(x-3) and (x−4)(x-4) respectively, find the values of a,b,ca,b,c. marks

[2021/RI/Prelim/I/1]

Solution

Q

Remainder and factor theorems

Given that the polynomial x4+ax2+bx+cx^4+ax^2+bx+c has a factor (x−2)(x-2) and gives remainders 12 and 26 when divided by (x−3)(x-3) and (x−4)(x-4) respectively, find the values of aa, bb and cc.

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Hint

  1. Let f(x)=x4+ax2+bx+cf(x)=x^4+ax^2+bx+c.

  2. A factor (x−2)(x-2) means f(2)=0f(2)=0. Remainders 12 and 26 mean f(3)=12f(3)=12 and f(4)=26f(4)=26.

  3. Three equations, three unknowns.

Answer · Q

a=−54,b=217,c=−234a=-54,b=217,c=-234.

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Three values of f, three planes in (a, b, c)

The unknown coefficients a, b, c are the axes.

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f passes through (2, 0), (3, 12) and (4, 26)

f(x) = x⁴ − 54x² + 217x − 234.

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  • Y = f(x)
  • Remainders
Ideas to try
The remainder on dividing by (x−k)(x-k) is just the height at x=kx=k. Where would you read the remainder for (x−1)(x-1)?